Q1: Does the moon orbit the earth, or does the moon orbit the sun?
I'm not convinced we should always answer directly. Very often
it is better to deflect the question and offer a better way of
thinking about the situation.
Rather than the qualitative question Q1, the better question is:
Q2: How can we understand and quantify what's going on?
That leads to the answer:
A2: /Separation of variables/.
The fact is, to an excellent approximation, the earth and
the moon orbit around their common center of mass ... and
then the pair orbits the sun. There are other things going
on, but they can be handled as small correction terms, e.g.
tidal effects.
This approach was used by Copernicus (1543) and Lagrange
(1764). Even Ptolemy (ca 160) used separation of variables,
with some degree of success.
..........
Even if there is a more precise qualitative question such as:
Q3: Is the Moon's orbit always concave toward the Sun?
then A2 remains the #1 method I would use to answer the question.
There are about a dozen ways of answering Q3, and the expert
should be able to use them all and cross-check them, but A2
is a good starting point, especially for students and other
non-experts. It has the advantage of answering Q3 and Q2
and Q1 and a lot of other questions besides. It gets used
again and again, in cosmology and subatomic physics and
everything in between.
Example: Normal modes of a compound oscillator ...
including standing waves in an organ pipe. This may
not look like the same problem, but the same method
contributes to the solution.
On the other hand, separation of variables is *not* guaranteed
to work. There are plenty of situations where it doesn't
work. Even when it does work, it is an approximation, and
you have to go back and check whether the approximation is
good or bad.
On the third hand, Q1 wasn't ever going to have an exact
answer anyway, so nobody should complain if we come up
with an approximate answer. No matter how you answer
Q1, you can replace that answer with A2 and get a better
model of what's going on.
It must be emphasized that very often you have to do a
change of variables before you can get a useful separation.
Students are unlikely to figure this out on their own.
Sometimes the required change of variables is tantamount
to a choice of reference frame. In this case, that's
tricky, because students are overly fond of attaching
their frame of reference to a material /thing/ such as
the earth of the sun. In contrast, attaching it to the
barycenter is not the first thing they are going to think
of. Let's be clear: The frame does not need to be
attached to a tangible thing ... and it does not need
to be personified in terms of an "observer". (This is
particularly important in special relativity, where the
"observer" is necessarily moving in the t direction,
but the reference frame is not.)
Seriously, in HS algebra when they introduced polar
coordinates, they did not require a special polar
"observer" or Cartesian "observer". The reference
frame exists just fine as an abstraction. I don't
know who introduced the idea of attaching "observers"
to frames, but it was a really bad idea.